Method apparatus and system for the identification of the relationship between two signals

ABSTRACT

A set of circuits with a common input and several outputs, wherein the second output signal is the time derivative of the first output signal, and the third output signal is the second time derivative of the first output signal, and wherein the transfer functions from the input signal to each output signal have the same denominator roots.

United States Patent 1191 Smith A I 111] 3,742,391 1451 June 26,1973

[ METHOD, APPARATUS AND SYSTEM FOR THE IDENTIFICATION OF THE RELATIONSHIP BETWEEN TWO SIGNALS [60] Continuation-impart of Ser. Nos. 826,085, May 15,

1969, Pat. No. 3,526,761, and Ser. No. Division of Ser. No. 826,085, May 15, 1969, Pat. No. 3,526,761.

1,638,437 8/1927 Gannet et al. 333/19 2,895,111 7/1959 Rothe 333/19 X 2,745,007 5/1956 Thomas 333/19 UX OTHER PUBLICATIONS Analysis of Linear Systems Cheng, Addison-Wesley Publishing Co., Reading, Mass, 1959, pages 175-178 and title page.

Primary ExaminerHerman Karl Saalbach Assistant Examiner-Marvin Nussbaum Attorney-Flehr, Hohbach, Test, Albritton & Herbert [57] ABSTRACT [52] US. Cl 333/6, 333/19, 333/20, 33 0 CR A set of circuits with a common lnput and several out- 51 1111. c1. H03h 7/06, H03k 5/00 Puts, wherein the Seeenel Output Signal is the time deriv- [58] Field of Search 307/160, 161; ative of the first Output Signal, and the third Output i 235/183; 333/6, 19, 20 nail is the second time derivative of the first output signal, and wherein the transfer functions from the input 5 References Cited signal to each output signal have the same denominator UNITED STATES PATENTS 2,859,914 11/1958 Balsingame 235/180 l3-Claims, 16 Drawing Figures Sign I 1 w 1 1 1 I 1 l 36 i i 37' I i l 1 l 1 l 1 1 1 1 1 1 1 1 1 1 l 1 1 1 1 l, J 1

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Patented June 26, 1973 3,142,391

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12 Sheets-Sheet 9 INVENTOR. Otto J M Smith Attorneys Patented June 26, 1973 3,742,391

12 Sheets-Sheet 1 0 INVENTOR Otto J M. Smith Ju W334 dl mzfflw Patgnted June 26,1973 3,742,391

12 Sheets-Sheet 11 FROM FIG. 5

Fig/5 FM I M INVENTOR. FP O 3 y OflO florneys azaWW 1 METHOD, APPARATUS AND SYSTEM FOR THE IDENTIFICATION OF THE RELATIONSHIP BETWEEN TWO SIGNALS CROSS-REFERENCE TO RELATED APPLICATION This application is a continuation-in-part and division of application Ser. No. 826,085, filed May 15, 1969 now US. Pat. No. 3,526,761, issued Sept. 1, 1970.

BACKGROUND OF THE INVENTION This invention relates to a method, apparatus and system for the identification of the relationship between two signals, and more particularly to a method, apparatus and system for the measurement of impedance and admittance functions, gain and phase, trans- LII fer functions. Fourier transforms of the time domain bridge arms, two terminals to which an unknown two-- terminal network can be connected, an oscillator to provide a single frequency, and a null detector. When the bridge is balanced, the impedance of the unknown twoterminal network can be read from the bridge dials as one complex number. This measurement is adequate only if the unknown contains only one energy storage element. If the unknown has several capacitors and inductors, the'measurements must be repeated at several frequencies. If as many measurements as unknowns are made, the solution for the unknowns can, be solved from a difficult set of equations. However, such solutions give very poor accuracy. To increase the accuracy, many more measurements can be made at many different frequencies but these must be combined statistically in an expensive high-speed digital computer. Another example arises in the measurement of the characteristics of amplifiers, servo mechanisms, three and four terminal networks, transmission lines, transducers, pneumatic and hydraulic information and power transmission devices, transistors, process controls, regulators, feedback control systems, and multivariate systems. One method has been to excite the input to a system with a frequency of known amplitude and to measure the output'amplitude and phase with respect to the'input. This measurement is repeated at many frequencies. Plots of gain and phase versus frequency, or a plot of gain versus phase are descriptions of the unknown'system, but to obtain the equation of this plot, it is difficult and has been done in the past by trail and error or by the fitting of templates. One method for measuring the characteristics of a network or system is to excite the system with an impulse or a step function and measure the impulse response or step response as a function of time. The analysis of such measured functions has been very difficult. Digital computers can be used to find a summation of exponentials that equals the measured time function or to find the Fourier transform of the measured time function. In the measuring of the characteristics of a random signal, it has been the practice to calculate the autocorrelation function of the signal. To utilize this function in system synthesis often required converting it the recorded-autocorrelation has to be multiplied by a sine and cosine wave and integrated over the entire function. It has been necessary to repeat this process many times for all significant frequencies. The results obtained are tabular or graphical in form and do not provide an equation of the power spectrum. From the foregoing, it can be seen that there is a need for a new and improved method, where and system for the identification of the relationship between two signals.

In general, it is an object of the present invention to provide a method, apparatus and system for identification of the relationship between two signals which overcomes the above named disadvantages.

Another object of the invention is to provide a method, apparatus and system of the above character which can be utilized for reading more than one complex number or more than two parameters of an unknown two-terminal network simultaneously.

Another object of the invention is to provide a method, apparatus and system of the above character in which more than one excitation frequency can be used simultaneously.

Another object of the invention is to provide a method, and apparatus of the above character in which gain and phase can be automatically measured utilizing only the signals existing in an operating system and without the introduction of a signal into the system.

Another object of the invention is to provide a method, and apparatus of the above character for determining the coefficients in the differential equation describing an unknown process. 1

into a power-density spectrum by taking the Fourier transform of the autocorrelation function. In general,

Additional objects and features of the invention will appear from the following description in which the preferred embodiments are set forth in detail in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a block diagram of the system and apparatus for the identification of the relationship between two signals incorporating my invention.

FIG. 2 is a circuit diagram, partially in block form, of one type of identification machine, apparatus or system incorporating my invention.

FIG. 3 shows a circuit diagram for typical state variable generators. I

FIG. 4 is a circuit diagram, partially in block form, of a computer for use in my identification machine.

FIG. 5 is a circuit diagram showing typical state variable networks for generating state variables.

FIG. 6 is a circuit diagram, partially in block form, of a computer similar to that shown in FIG. 4.

FIGS. 7 and 8 are circuits which receive excitation signals from FIG. 9 and generate three excitation state variables and multiplies each by its corresponding state FIG. 12 is a network for generating state variables which can be used in place of the network shown in FIG. 7.

FIG. 13 is a circuit diagram ofa state variable generator.

FIG. 14 is a network consisting of a combination of state variable computers and coefficient multipliers.

FIG. 15 is a state variable transformation analog computer to be used in conjunction with the embodiments shown in FIGS. 3 and 4.

FIG. 16 is a state variable transformation analog computer also to be used in conjunction with the embodiments shown in FIGS. 3 and 4.

In general, the present invention for determining the relationship between two signals is characterized by the provision of novel means for generating state variables from the two signals which are functions of time and the provision of novel means for forming a trail differential equation relating the two signals. Means is provided for forming an error measure of the error between the trail differential equation and the actual differential equation, minimizing the error measure by summing the weighted state variables, and using parametric feedback to correct the trial differential equation until the sum of the weighted state variables is minimized.

More specifically, the apparatus and system utilized for identifying the relationship between two signals comprises means for generating a plurality of first derived signals linearly related to the first of the two signals and means for generating a plurality of second signals linearly related to the second of the two signals. Means is also provided for determining a plurality of coefficients and multiplying the same with the first and second derived signals. Means is provided for forming a sum of the products and amplifying the same. Means is provided for generating a signal containing a component related to the product of the amplified sum and one of the state variables. Means is also provided for adding the last named signal to the predetermined coefficient which was previously multiplied as a factor times said one of the state variables.

More in particular, there is shown in FIG. 1 a system and apparatus for identifying the relationship between two signals. This apparatus may also be called an identification machine. This apparatus or machine consists of a signal generator 21 which delivers an excitation signal 22 to excite an unknown process 23. The output from the signal generator 21 is also supplied on a circuit 24 to an excitation state variable network 26 which produces a plurality of excitation state variables on a plurality of output circuits 27 which are linearly related to the excitation signal from the signal generator 21. The response of the unknown process 23 to the excitation signal from the signal generator 21 is supplied on an output circuit 28 to a response state variable network 29 which produces a plurality of response state variables on a plurality of output circuits 3] which are linearly related to the output signal from the unknown process 23.

In general, it can be stated that the excitation state variables on the circuits 27 are related to one another by the mathematical operation of either integration or differentation, and similarly the response state variables on the circuits 31 are related to one another by the mathematical operation of integration or differentation. The state variable networks 26 and 29, are hereinafter explained, can each consist of one network with a plurality of taps so that different voltages are available or can consist of a plurality of different networks, each one of which produces a separate voltage. The networks 26 and 29 can be of any suitable type so long as a linear relationship is created between the input to the network and the output of the network. Preferably, the relationship between the state variable and the excitation signal supplied to the network are such that the input signal has a denominator term in the transfer function which is the same as the denominator term in the transfer function for the relationship between each state variable and the input to the network or between each response state variable and the response signal.

As can be seen in FIG. 1, the excitation state variables on the circuits 27 and the response state variables on the circuits 31 are supplied to a computer 32. The purpose of the computer 32 is to form a trial differential equation. The differential equation consists of a constant times one of the state variables plus a different constant times a different one of the state variables, etc., so that a sum of all of the constants each times its respective state variable is equal to zero. If the constants are all of the proper values, the sum of the equation will be zero and the constants will properly represent the differential equation of the unknown process 23. However, if the constants are of the wrong values, then the sum of the products of each constant times its corresponding state variable will add up to provide an error function which is different than zero. The computer 32 analyzes the error function to bring it to zero by determining which constant is in error and in which direction it is in error. This analysis of the decomposition of the error function into its components due to the errors in the different constants is performed by multiplying the error function times one of the state variables. This product will have an average essentially equal to zero if the constant corresponding to that state variable is correct; an conversely, if the constant corresponding to the state variable is wrong, then this product will have a non-zero mean or an average value which has a magnitude proportional to the error in the constant and a polarity proportional to the polarity of the error in the constnt. To correct the constant, then, the product of the state variable times the error function is integrated and the integral is added to the constant corresponding to that state variable. The polarity of the feedback loop is chosen to reduce the error to a minimum. With a feedback loop on all but one of the constants associated with the excitation and response state variables, and each feedback loop arranged to minimize the error due to the constant which it is controlling, the error function will be driven to an average of zero.

The useful information which this computer 32 derives is the set of constants which are changed until the sum of the products of these constants times the corresponding state variables is continuously equal to zero. This useful information is read out by a display means 33. For example, if the constants are obtained as the output voltages of a bank of integrators, then a meter can be switched to any one of the integrators to read its output voltage. The unknown process 23, as shown in FIG. 1, is intended to represent any unknown device which can receive an excitation and have a response which is related to the excitation by a linear differential equation. For example, the process could be an audio amplifier in which the excitation is the microphone input and the response is the loudspeaker output, or the process could be a dynamo electric amplifier in which the excitation is a field voltage and the reponse is an armature voltage, or the process could be a two-terminal filter in which the excitation is the voltage impressed across the two terminals and the response is the current which flows into one terminal and out of the other. Or, the process could be a hydraulic transmission system in which the excitation is the displacement of a hydraulic valve and the response is the force on a hydraulic cylinder. 1

The signal generator 21 in FIG. 1 should not be a fixed single frequency oscillator. Preferably, it is a random noise or signal generator such as a'low frequency gaussian noise generator manufactured by Automation Laboratories, Inc. of 179 Liberty Ave., Mineola, L. I., N. Y.; but it may be a square-wave generator, or a triangle wave generator, or a repetitive pulse generator, or a random pulse generator. The signal generator 21 in FIG. 1 may be a sweeping oscillator or an FM modulated oscillator whose rate of change of frequency is very large compared to the rate of convergence of the computer 32 in FIG. 1 to the values of the constants. The reason for this is that during the time of convergence, a large number of different frequencies should have passed through the process. The reason that the signal generator should vary its frequency rapidly is that it must-deliver'a wide variety of different frequencies for the excitation of the process during the time that the computer is changing the constants which it is evaluating. Y

The coefficients of the differential equation read out by the display means 33 can be designated as a a,, a .a,, and b b b b Within the computer 32, there is provided means for forming the sum'e of the products of the state variables times the corresponding coefficients. The state variables on the circuits '27 can be designated as X X,, X The state variables on the circuits 3] can be designated as Y Y Y The equation'for the sum e can be written as follows:

'The computer 32- is also provided with means for forming the plurality of products of each state variable times the function of the sum e. If the function of the sum e is designated f(e), then this plurality of products cfl fl nfl kfl qfl rf( Y,,f(e), Y f(e). The computer 32 is also provided with means for integrating each of the above products and changing each of the coefficients but one in response to a time integration of one of the products in the plurality of products above. Specifically, the time integrations are:

The meaning of the integral notation used above with time limits from minus infinity to zero is that the integration has been carried out from the time when the equipment was turned on until the present time. No ini tial value of the coefficient is shown in each equation above because in normal operation the integration is continued for an amount of time sufficient to destroy the initial values of the coefficients at the time that the equipment was turned on.

With the proper choise of the function f(e),e will be driven towards a minimum, and the coefficients a and b will change due to the action of the integrators until they reach final steady state values, which values will satisfy the following differential equation for the unknown process 23 giving the relationship between the excitation on circuit 22 called X and the response on circuit 28 called Y.

The function f(e) can be linearly proportional to e, such as would be obtained from an amplifier whose input is e. Alternatively, the function f(e) can be the polarity of e only, i.e., e/ IQI. In this case, the function can be generated by the motion of the armature of a relay whose coil current is driven by the output of an amplifier whose input is e. The plurality of products above can then be obtained by connecting each state variable to reversing contacts mounted on the armature of the relay.

It should be appreciated that the present invention is not limited to the function f(e) enumerated above. For example, the function e [u] may be used, or the function 2 may be used. As another axample, for m equal to 2 or greater, the function may be FIG. 2 shows a wiring diagram partially in block form of an identification apparatus ormachine of the type shown in FIG. I-. Analog computer notation is used in FIG. 2 and analog computer terminology will be used in describing the operation of FIG. 2. The signal X to the unknown process 23 is derived from the signal generator 2las explained in FIG. 1. The signal Y is the response from the unknown process 23. Signal Xpsses on circuit 24 through two different filter networks 36 and 37 which make up the state variable network 26 and generate two different voltages X o and X respectively. The voltage X is generated by the network 36 which is a lag filter formed by a series resistor R and a shunt capacitor C so that the output voltage is read across the capacitor and is related to the signal X by the transfer function 1/1 sT. The signal X is produced by the filter network 37 which consists of a series capacitor C and a shunt resistor R so that the voltage is read across the resistor R. The signal X, is related by the transfer function sT/l sT so that the signal X is the pure derivative of the signal X times the constant T. The signals X and X, correspond to the excitation state variables appearing on the circuits 27 in FIG. 1.

In a similar manner, the output signal Y is supplied to a pair of filter networks 38 and 39 which form the response state variable network 29 to produce two different voltages Y and Y which correspond to the state variables appearing on the circuits 31 of FIG. 1. The signal I is produced by a lag filter network 38 which is formed in the same manner as filter network 36 so that the output signal Y is related by the transfer function l/l+sT tov the input signal Y. Similarly, the output signal Y is produced by a lead filter network 39 so that the output signal y is related by the transfer function sTll+sT to the input signal I. More generally stated, the state variable networks 26 and 29 are preferably chosen so that the network producing the state variable X is identical to the network producing the state variable Y The same is true for X and Y,.

The state variables X X Y and I, are supplied to the computer 32 which, in the embodiment shown in FIG. 2, consists of a stepping switch 41 which is provided with two banks 42 and 43 of stationary contacts adapted to be engaged by a pair of movable contacts 44 and 46, respectively. The state variable signalsare connected to the stationary contacts of bank 42 so taht one of the state variables can be selected at a time to be supplied to the input of a multiplier 48 which also can be identified as an analyzer. The contacts of bank 42 are, therefore, identified as X X,, Y and Y The second bank of stationary contacts 43 of the stepping switch 41 are identified'as d d b an 5, which are adapted to be contacted by the movable contact 46 and to be supplied with energy from the analyzer 48.

The feedback loop 55 shown in FIG. 2 when the contact 46 is in engagement with the 0 contact will produce continuously a rate of change of a in the correct direction until the error function entering theanalyzer multiplier 48 is minimized. The normal operation of the apparatus shown in FIG. 2 is not to permit the constant a to be adjusted for a sufficient length of time to minimize the error function but only to permit the constant to change a small amount and then the stepping switch steps to the next contact and receives the state varable X analyzes the error function utilizing the state variable X and uses this analysis to correct the constant a,, and making only a small part of the total correction necessary in the constant a,. Then, the stepping switch continues to step through the other contacts and when it has cycled through all positions, it returns to the position as shown in FIG. 2 and makes an additional correction in the constant 0 and repeats this cycling in a periodic manner until the necessary correction is obtained for all of the constants In accordance with conventional analog computer notation, all of the multipliers and the summing amplifiers and the integrators in FIG. 2 are assumed to be of the inverting type such that the output is the negative of the operation on the input which the device is intended to perform.

A clamping switch 61 is provided which is connected to the parameter b and holds it at the constant value of +1 irrespecive of the output of the integrator connected to b The purpose for this clamping switch 61 is so that a differential equation utilizing two terms in the numerator and two terms in the denominator has four coefficients but only three of the coefficients are independent, that is, one can select any one coefficient and divide all the others by it and obtain a correct differential equation. One whould simply have changed all the numerator terms and all the denominator terms up or down by some constant factor. In order to set the scale of these factors and have only as many converging operations in the identification machine as the number of independent variables, one can, therefore, solve only three of the four coefficients. By setting the parameter b equal to one, then the other three coefficients can be solved for. This is satisfactory if the unknown process contains either differentation or gain at zero frequency but this is not satisfactory if the unknown process contains pure integration. In that case, the parameter b should be zero. With the clamping switch 61 in the position shown in FIG. 2, the parameter b is held at unity and this will cause all the other parameters to tend to increase to very large numbers. For measuring an unknown pr'ocess containing pure integration, the clamping switch 61 of FIG. 2 is thrown to the upper position shown in FIG. 2 in which the parameter a is held at unity. Then, during the normal operation of the machine, the parameter b will reach a finite number and the parameter b will reach zero. Thus, when the clamping switch 61 is in the upper position, the machine is satisfactory for measuring unknown processes which have either unity d-c gain or infinite d-c gain due to one or more integrations.

The state variable networks shown in FIG. 2 are of a unique type. They are chosen such that the network producing the state variable X is identical to the network producing the state variable Y The reason for this is that the poles of this state variable network are, therefore, removed from the diferential equation and permit a representation of the unknown process by a set of parameters closely related to the coefficients of the conventional differential equation. In 5 similar manner, the poles of the network to produce the state variable X l are the same as the poles of the network to produce the state variable Y,. This also results in a simplification of the interpretation of the parameters which are obtained by the identification procedure and also result in a larger number of independent variables which can be evaluated by the identification machine for a given quantity of equipment. In addition, in FIG. 2, a further improvement has been made by setting the poles of the state variable network to produce the state variable X also equal to the poles of the state variable network to produce the state variable X In other words, not only do the state variable networks appear in pairs which are identical, -i.e., the pair for X Y and the pair for X I, which is one important requirement, but, in addition, the poles of one pair are equal to the poles of the other pair. This, then causes these pole terms to completely cancel out of the relationship which is fulfilled by the identification machine so that the relationship which is fulfilled by the identification machine has the same parameters in it as the conventional differential equation, i.e., the parameters a (1,, b and 11 are the coefficients of the first order differential equation representing the unknown process. If the unknown process contains second-or third order terms, then the identification machine must contain additional state variables to analyze for these higher order terms. In general, the state variable machine will contain many of these state variable networks and many parameters like a, and 19,, but for the purposes of illustration, FIG. 2 has been shown sufficient to identify in the unknown process one numerator zero, one denominator pole and one gain term.

Previous investigators who have attempted to build identification machines of this type have tried to generate the state variable X, by calculating the derivative of the signal X and have tried to generate corresponding state variable X by generating the second derivative of the signal X. Now it is well known to those skilled in the art that pure first derivatives and pure second derivatives cannot in fact be calculated, and consequently, previous attempts to build identification machines have resulted in state variables with errors that are related to the dynamic mistakes made in attempting to generate state variables with unrealizable networks.

The unique networks used in FIG. 2 are realizable networks as hereinafter described such that the derivative is generated with its corresponding pole which cannot be eliminated and then the state variable X is generated by using the pole alone. This produces a relationship between state variables which contain no error, that is, the state variable X, is the pure derivative of the statevariable X 0 and nothing need be said about its relationship to the signal X. Or stated another way, the state variable X, is not thederivative of the signal X. It is the derivative of the signal X with a significant amount of filtering. Computer 32 as shown in FIG. 2 is more fully described in U.S. Pat. No. 3,526,761.

From the foregoing, it can be seen that the network 26 comprises means for generating a plurality of first derived signals which are linearly related to the first of two signals, that is, the input signal X. The network 29 consists of means for generating a plurality of second derived signals which are linearly related to the second of two signals, that is, the output signal Y, from the unknown process. The coefficients a,, (1, b,, b which are determined can be called weighting factors. These weighting factors are multiplied by multipliers 51 with the corresponding linearly derived signal. The products obtained are added in the summing amplifier 52.

In FIG. 3, there is shown circuitry for typical state variable generators 26 and 29. The input to the unknown system or proces 23 is designated by X and the output from the unknown system is designated by I. As shown in FIG. 3, the input signal X is decomposed into the derived state variables designated by X,,, X,, X X and It through an RC network, and similarly the negatives of these same state variables are derived through the computer 32 whichreceives the plurality of state variables generated by the circuitry as shown in FIG. 3. The state variables are connected to the terminals indicated in FIG. 4. These state variables are supplied to relay 89. When 0, is a positive, the relay 89 is operated vthe use of an inverting amplifier indicated by 'X so that the movable contacts of the reversing switches 84 engage the upper contacts, and conversely when 0,

is negative, the movable contacts engage the lower contacts.

Operation of the computer shown in FIG. 4 as a part of the identification machine may be briefly described as follows. Let h, and k, be the machine outputs. The machine forms the trial integro-differential equation:

it 2 i i i i) c i=0 ((1) The purpose of the identification machine is to find the unknown differential equation representing the true relationship between the state variables, which is and k, which relate the state variables of the unknown system in the manner shown by equation 7, when it has been running long enough, so that the value 6,, from equation 7, or the value of the input to the relay shown in FIG. 6, is zero continuously in time. Any error in a coefficient h, or k, will cause the variable t9 to deviate from zero during transients and during the progress of random variations of the state variables.

In FIG. 4, each coefficient will converge to its correct value because of the following mechanism: Assume that all of the coefficients are correct except h,; assume further that X, is positive and that X, is negative; then the output of the multiplier connected to h, will be too negative if h, is too positive or will be too positive if h, is too negative. This output will contribute an error component to 0, such that the relay will move to the top contact if h, is too negative and the relay will move to the bottom contact if h, is too positive. With the relay on the top contact the integral connected to the state variable X, will integrate a positive number and will therefore increase the variable h, bringing it from its too negative value towards its correct value. In a similar manner when the relay is on its bottom contact, 9 the input to the integrator for h, will integrate a negative number and drive h, from its too positive value again towards its correct value. This negative feedback iterative convergence of the assumed value of h, in the machine towards the correct value as determined by the relationship between the state variables X, and Y, will always have the correct direction of change for either polarity of X In a similar manner, each h, and k, in FIG. 4 will converge towards its correct value independently of the polarity of the associated state variable. The rate of convergence of each coefficient is proportional to the magnitude of the associated state variable. For example, if state variable Y should be always very small, then the rate of convergence of k will always be small. Therefore, those coefficients that are most important in the integro-differential equation will assume their correct values most rapidly, and those coefficients that are least important will take the longest time before their values become significant. When the identification machine is first turned on, it calculates a first order linear approximation to the unknown process, and then gradually improves upon this by calculating a second order approximation and then a third order approximation and continuously expands the complexity of the model used to represent the unknown process. The accuracy is proportional to the running time. If all of the state variables are stationary random variables, and the engineer does not know how much data is necessary in order to obtain a valid model of the unknownprocess, then the identification machine can be operated in real time until the rates of change of the identified parameters are negligible in the judgment of the engineer. There is no need to record the input and output signals from the unknown process.

Note that one parameter, h has been chosen arbitrarily in FIG. 4. This sets the scale factor for the other parameters. For example, if the unknown process has a dc gain of 100, then k will equal 001 for the usual designation of state variables. If the unknown process has an integral, then k will equal 0 in FIG. 4.

The computer shown in FIG. 4 will work satisfactorily for a wide variety of selections of state variables. A convenient selection of state variables is shown in FIG. 3. In this case, the input to the unknown system is the variable X. The state variable X is chosen as the random variable X filtered with a four-time-constant lowpass filter. This provides a desirable smoothing of the fluctuations, particularly for those coefficients calculating the low pass characteristics of the unknown system. The state variable X, is related to the input X by the same four-time-constant filtering network, but in addition, there is one zero factor in the numerator of the transfer function from X to X,. X has a second order numerator term and the same four-time-constant filtering poles. X has a third order numerator term. All of the resistors have the same value R, and all of the capacitors have the same value C. The transfer functions relating this particular selection of state variables to the random input X are p" =d"/dt" T= RC D =1+ 10 pT+ (pT) 70111) (p X X/D X, 1 +pT) X/D x, 1 3 pT+ (pT) X/D X3 1 6 pT+ 5 r W) X/D In a similar manner I, state variables are related to the random output Y by the following equations The identification machine solves for the values of h; and k in equation 7. If equations 1 1 20 inclusive are substutited into equation 7, one has the following The usual method of writing the differential equations for the unknown system in FIG. 3 in terms of the random input X and the random output Y would be 1 i i g? (my; (17) Y) (23) This is a shorthand method of writing a X a pX a PX a p X a P X b b pY b P Y b P Y b.pY O

It can be seen from a comparison of equation 22 and equation 24, term by term, that the parameter transformation between the conventional differential equation notation and the transformed state variables, which are used in FIG. 3, is

From the foregoing, it can be seen that the identification machine, apparatus or system identifies an integrodifferential equation. Since the identification machine is connected tothe input and the output of the unknown system, it observes these signals as they vary with time. There is no need to perturb the input when the unknown system is operating in its usual manner, and there exists commands and other variations in the input signal.

The differential equation is formed as an integrodifferential operation on the input variable plus a different integro-differential operation on the output variable summing to zero. The identifier starts with assumed integro-differential parameters or coefficients and forms the function which should equal zero. Errors are represented by-deviations from zero, and these are used to slowly correct each of the parameters in the assumed integro-differential operations.-

The convergencerate of the machine is adjustable and can be chosen in accordance with the spectra of the input and output signals. For a variable-parameter system, if the parameter variation is slow, the identification machine will track the parameter variations and will deliver the value of the parameter as a function of time. For a variable-parameter system where ,the parameter variation is quite rapid or where the parameter is a function of one of the signals in a non-linear manner, then the identification machine delivers a statistical' best estimate of the linearized parameter, smoothing out the variations.

In FIG. 5, there is shown another type of circuitry for generating state variables in which the state variables are closely related to the derivatives of the input and output of the unknown system. By comparisonof FIGS. 3 and 5 with FIGS. 2, 12, 13 and 14, it can be seen that the embodiments in FIG. 3 and 5 are similar to those shown in FlGS. 2, 12, 13 and 14 with the exception that in FIGS. 3 and 5 the differentiations are performed by bilateral loaded networks rather than by isolated diffen entiators as-in FIGS. 2, 16, 19 and 22. The four energy storage networks shown in FIG. 5 will have denominatorsof their transfer functions of fourth order. The state variable X, is, therefore, the fourth derivative of the random variable X, but with an additional filtering produced by four poles. This additional filtering is not undesirable. The state variable X 1 has a composite transference with a derivative filtered by three zeros and four poles. The equations for all of the state variables in FIG. 5 are given by D 1 7(pT)+15 (pT) 10(pT) (pT) X0 E X (36) X. (p 5w w) on) X/D X. (PT)2 on on) X/D X. E on on) X/D X. or) X/D Y1 (PT (p 6(PT)3 (PTV) Y/D Y. (m 01 m) Y/D Y. on on) Y/D (m Y/D These state variables are inserted into the integrodifferential equation 6 for the identification machine, and the entire equation operated on by the operator D from equation 35. Then it can be seen that the parameter transformation for FIG. 7 is 

1. A multiport linear passive lumped-parameter network consisting of only linear passive elements with one input port and at least four output ports, means for providing an input signal at said input port, passive coupling means within said network for producing four output signals at the four output ports, each frequency component of said signals being described by an envelope magnitude and phase the frequency response of each of said output signals with respect to said input signal being finite (including zero) at all real frequencies, the second output signal having a phase leading the phase of the first output signal by 90* and the second output signal envelope magnitude being proportional to W times the first output signal of radian frequency W, the third output signal for each input signal having a phase leading the phase of the first output signal by 180 degrees and the third output signal envelope magnitude being proportional to omega 2 times the first output signal envelope magnitude, the fourth output signal for each input signal having a phase leading the phase of the first output signal by 270* and the fourth output signal envelope magnitude being proportional to omega 3 times the first output signal envelope magnitude for each input signal of radian frequency omega .
 2. A multiport network consisting of only linear passive elements with one input port and three output ports, means for providing a voltage excitation signal having a parameter T at said input port, coupling means within the network to generate first, second, and third output signals responsive to said excitation at the three output portS respectively, such that when the excitation (after a preselected time t equals zero) produces a first output signal changing in magnitude proportional to s1 (t) T2 (1 - (1 + t/T) e t/t) then the corresponding second output signal for positive time is proportional to s2(t) t e116 t/T and the corresponding third output signal for positive time is proportional to s3(t) (1 - t/T(e t/T for all T
 3. A claim as in claim 2 wherein s1(t) is proportional to the time integral of s2(t) and s2(t) is proportional to the time integral of s3(t).
 4. A claim as in claim 2 wherein s2(t) is proportional to the time derivative of s1(t) and s3(t) is proportional to the time derivative of s2(t).
 5. A linear passive multiport circuit consisting of only linear passive elements comprising a two-terminal input with a source of voltage supplies to said input, a first pair of two impedances with different phase angles in series between the said two terminals with a first tap between the said two impedances, a second pair of two impedances with different phase angles in series between said first tap and one of said input terminals, with a second tap between the two said impedances of said second pair, a third pair of two impedances with different phase angles in series between said second tap and one of said input terminals, with a third tap between the two said impedances of said third pair, passive said first pair of two impedances generating a second signal proportional to the voltage between said first tap and said one input terminal, said second pair of two impedances generating a third signal proportional to the voltage between said second tap and said one input terminal, said third pair of two impedances generating a fourth signal proportional to the voltage between said third tap and said one input terminal, and passive means responsive to an algebraically linearly weighted combination of said, second, third and fourth signals to generate a fifth signal.
 6. A circuit as in claim 5 whereby said third signal is proportional to the time derivative of said second signal, and said fourth signal is proportional to the time derivative of said third signal.
 7. A circuit as in claim 5 whereby said third signal is proportional to the time integral of said second signal, and said fourth signal is proportional to the time intergral of said third signal.
 8. A linear passive circuit consisting of onlp linear passive resistors and capacitors for producing a plurality of state variable signals from a measured signal, comprising means for performing a dynamic passive operation on said measured signal to produce a first output, means for performing a second passive dynamic operation on said measured signal to provide a second output, means for performing a third passive dynamic operation on said measured signal to provide a third output, means for performing a fourth passive dynamic operation on said measured signal to provide a fourth output, whereby said second output is the derivative with respect to time of said first output and said third output is the time derivative of said second output, and said fourth output is the time derivative of said third output, and whereby the roots of the denominators of the transferences from said measured signal to each of said outputs are equal
 9. A multiport circuit consisting of linear passive resistors and capacitors for producing a plurality of state variable signals from an input signal, passive means for generating from said input signal a first state variable signal, passive means for generating from said input signal a second state variable signal which is a time derivative of the first state variable signal, and means For generating from said input signal a third state variable signal which is a time derivative of the second state variable signal, and passive means for generating from said input signal a fourth state variable signal which is a time derivative of the third state variable signal, each of said means for generating a state variable signal having a gain less than a predetermined finite value of gain, the roots of the denominator of the complex gain as a function of frequency from said input signal to said first state variable signal being equal to the roots of the denominator of the complex gain from said input signal to said second state variable signal and being equal to the roots of the denominator of the complex gain from said input signal to said third state variable signal and being equal to the roots of the denominator of the complex gain from said input signal to said fourth state variable signal.
 10. A linear passive dynamic circuit consisting of only linear passive elements for producing a plurality of state variable signals from an input signal, passive means for generating a first state variable signal from said input signal, passive means for generating second, third, and fourth state variable signals from said input signal, said state variable signals being related such that a polynomial in the frequency variable times the Laplace transform of the first state variable signal using the frequency variable equals the Laplace transform of said input signal and so that the fourth state variable signal is the time integral of the third state variable and so that the third state variable signal is the time integral of the second state variable signal and the second state variable signal is the time integral of the first state variable signal, each of said passive means for generating a state variable signal having a gain from input to state variable signal less than a predetermined value.
 11. A circuit as in claim 8 wherein said transference from measured signal to the said third output has a numerator polynomial order not greater than its denominator polynomial order.
 12. A circuit as in claim 9 wherein each means for generating a state variable signal contains an R-C ladder network.
 13. A circuit as in claim 10 wherein each of said gains is limited to less than said predetermined value only by the linear dynamic properties of each of said means for generating a state variable signal. 